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70                       ANALYTICAL MECHANICS
equation and passing from the logarithmic; to the exponential form, we have                  _^
J.  — C6      j
where c is the constant of integration. If 0 is measured from the normal to the surface at the point whew the right ~ha,nd side of the cord leaves contact we obtain the init ial eondil ion, T = To when 0 = 0, which determines c. Applying this condition to the last equation we have
T=TQe^°.                                (4)
DISCUSSION. — Equation (4) gives the relation between the values of the tensile force at any two points of the cord. It must be observed that 9 is measured in the same direction as F; in other words, opposite the. direction towards which the cord is urged to move. Therefore T or 7'0 has the larger value according to whether 0 is positive or negative. As a concrete example suppose a weight W to be suspended from the right-hand end of the cord and to be held in equilibrium by a force /'' applied at the left-hand end. If F is just large enough to prevent H" from falling then the cord will be on the point of moving to the right, therefore 0 is measured in the counter-clock wise T direction and is positive. In this case
F = We"*9.
In case F is just large enough to start Wio move up, then 0 is measured in the clockwise direction and is negative. Therefore
The value of T drops very rapidly with the increase of 6.    This fact „ is made clear by drawing the graph ^ of equation (4), Fig, 45.   The graph
Pio. 4f>,
may be constructed easily by making use of the half-value period curve.  If P denotes the period, then, by definition, the ordinal** is rei lu to one-half its value every time P m added to 0.*    Wo have therefore
f th
* The difference between this definition of /* and the ow given in tin* preceding section is accounted for by the <linVrew<t in the Mi^n* of th«< expciru'nts in equation (4) and in equation (14) of the pn*c-eding station.